Finding the right questions to increase accuracy in classification
The problem you're trying to address can, in some sense, be viewed as a Feature Selection problem. If you look for literature using only those words, you're not going to find what you're looking for though. In general, "Feature Selection" simply refers to the problem where you already have a large amount of features, and you're simply deciding to select which ones to keep and which ones to throw away (because they're not informative or you don't have the processing power to try training with all features for example).

I'd recommend looking around for a combination of "Feature Selection" and "Cost-Sensitive". This is because, in your case, there are costs associated with selecting features; values may be costly to obtain for some features. Searching for this combination leads to publications which look to be interesting for you, such as:


Cost-sensitive feature selection using random forest: Selecting low-cost subsets of informative features
Cost-sensitive Dynamic Feature Selection
Cost-Effective Feature Selection and
Ordering for Personalized Energy Estimates
and probably much more...


I cannot personally vouch for any of those techniques since I've never used them, but those papers certainly look relevant for your problem.



When you're looking around for more literature, terms like "cost", "cost-based", maybe "budgeted" are crucial to include. If you don't include those, you're just going to get papers on problems like:


Feature Selection: given a set of features/columns, which ones am I going to use across all samples/instances/rows?
Feature Extraction: given data (typically without clear human-defined features, like images, sound, etc.), how am I going to extract relevant features from this?
Active Learning: given a bunch of samples without labels but feature values already assigned, which one would I like an oracle/human expert/etc. to have a look at so that they can tell me what the true label is?


Those kinds of problems all do not really appear to be relevant in your case. Active Learning may be somewhat interesting in that it is about trying to figure out which rows would be valuable to learn from, whereas your problem is about which columns would be valuable to learn from. There does seem to be a connection there, Active Learning techniques might to some extent be able to inspire techniques for your problem, but just that; inspire, they likely won't be 100% directly applicable without additional work.